Stability of the minimal surface system and convexity of area functional

Abstract:

We study the convexity of the area functional for the graphs of maps with respect to the singular values of their differentials. Suppose that $f$ is a solution to the Dirichlet problem for the minimal surface system and the area functional is convex at $f$. Then the graph of $f$ is stable. New criteria for the stability of minimal graphs in any co-dimension are derived in the paper by this method. Our results in particular generalize the co-dimension one case, and improve the condition in the 2003 paper of the first author and M.-T. Wang from $|\bigwedge ^{2} df|\leq \frac {1}{p-1}$ to $|\bigwedge ^{2} df|\leq \frac {1}{\sqrt {p-1}},$ where $p$ is an upper bound of the rank of $df$, and the condition in the 2008 paper of the first author and M.-T. Wang from $\sqrt {det(I+(df)^Tdf)} \leq \frac {43}{40}$ to $\sqrt {det(I+(df)^Tdf)} \leq 2$.